/*
  This code can be loaded, or copied and pasted, into Magma.
  It will load the data associated to the HMF, including
  the field, level, and Hecke and Atkin-Lehner eigenvalue data.
  At the *bottom* of the file, there is code to recreate the
  Hilbert modular form in Magma, by creating the HMF space
  and cutting out the corresponding Hecke irreducible subspace.
  From there, you can ask for more eigenvalues or modify as desired.
  It is commented out, as this computation may be lengthy.
*/

P<x> := PolynomialRing(Rationals());
g := P![3, 2, -5, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {13, 13, -w + 2}>;

primesArray := [
[3, 3, w],
[4, 2, -w^3 + 3*w^2 + w - 2],
[11, 11, -w^3 + 3*w^2 + 2*w - 5],
[13, 13, w^3 - 2*w^2 - 4*w + 2],
[13, 13, -w + 2],
[17, 17, w^3 - 3*w^2 - 2*w + 2],
[19, 19, -w^2 + w + 4],
[19, 19, w^3 - 2*w^2 - 3*w + 2],
[23, 23, w^2 - 2*w - 1],
[27, 3, w^3 - 2*w^2 - 5*w - 1],
[29, 29, -w^3 + 2*w^2 + 5*w - 1],
[37, 37, -w^3 + 2*w^2 + 5*w - 4],
[41, 41, 2*w^3 - 5*w^2 - 6*w + 4],
[53, 53, w^3 - 2*w^2 - 3*w - 2],
[59, 59, w - 4],
[67, 67, 2*w^2 - 3*w - 8],
[73, 73, 2*w^3 - 5*w^2 - 6*w + 5],
[89, 89, -2*w^3 + 6*w^2 + 5*w - 7],
[97, 97, -2*w^3 + 6*w^2 + 3*w - 7],
[97, 97, -w^3 + 3*w^2 + 3*w - 1],
[103, 103, w^2 - 4*w - 4],
[113, 113, 2*w^3 - 5*w^2 - 4*w + 1],
[113, 113, -w^3 + 3*w^2 + w - 5],
[139, 139, 2*w^2 - 3*w - 4],
[139, 139, w^3 - w^2 - 6*w - 1],
[149, 149, 2*w^3 - 5*w^2 - 5*w + 2],
[149, 149, w^3 - 4*w^2 + 10],
[151, 151, -3*w - 1],
[157, 157, w^2 - w - 8],
[157, 157, w^3 - 7*w - 8],
[163, 163, -3*w^3 + 8*w^2 + 8*w - 10],
[163, 163, w^3 - 3*w^2 - 4*w + 7],
[163, 163, -w^3 + 4*w^2 + w - 8],
[163, 163, -2*w^3 + 5*w^2 + 7*w - 4],
[167, 167, -2*w^3 + 5*w^2 + 8*w - 4],
[169, 13, -2*w^3 + 4*w^2 + 7*w - 5],
[173, 173, 2*w^3 - 4*w^2 - 9*w + 4],
[173, 173, -w^3 + 5*w^2 - 2*w - 5],
[181, 181, 3*w^3 - 7*w^2 - 12*w + 7],
[191, 191, w^3 - 3*w^2 - 3*w - 1],
[199, 199, 2*w^3 - 3*w^2 - 10*w - 1],
[199, 199, 2*w^3 - 6*w^2 - 6*w + 13],
[223, 223, -2*w^3 + 6*w^2 + 4*w - 11],
[223, 223, 3*w^3 - 7*w^2 - 12*w + 8],
[223, 223, -2*w^3 + 5*w^2 + 6*w - 1],
[229, 229, -w^3 + 3*w^2 + 2*w + 1],
[229, 229, 3*w^3 - 4*w^2 - 15*w - 5],
[233, 233, -3*w^3 + 7*w^2 + 11*w - 5],
[233, 233, 2*w^3 - 5*w^2 - 4*w + 5],
[241, 241, w^3 - 3*w^2 - 5*w + 7],
[257, 257, w^3 - 3*w^2 - 4*w + 8],
[263, 263, w^2 - 4*w - 5],
[271, 271, -2*w^3 + 6*w^2 + 5*w - 13],
[277, 277, -2*w^3 + 6*w^2 + 3*w - 8],
[277, 277, w^3 - 2*w^2 - 2*w - 2],
[281, 281, 2*w^3 - 5*w^2 - 4*w + 2],
[283, 283, -3*w^2 + 7*w + 10],
[283, 283, -w^3 + 4*w^2 - w - 7],
[293, 293, 2*w^3 - 5*w^2 - 3*w + 4],
[293, 293, w^3 - 8*w - 4],
[307, 307, -w^3 + 2*w^2 + 6*w - 2],
[311, 311, w^2 - 5],
[317, 317, -w^3 + 5*w^2 - 2*w - 11],
[331, 331, -w^3 + 3*w^2 - 5],
[349, 349, -2*w^2 + 5*w + 2],
[353, 353, -2*w^3 + 5*w^2 + 4*w - 4],
[353, 353, 3*w^3 - 7*w^2 - 10*w + 2],
[353, 353, -2*w^3 + 7*w^2 + 2*w - 10],
[353, 353, w^3 + w^2 - 9*w - 13],
[361, 19, -w^3 + w^2 + 5*w - 1],
[373, 373, -w^3 + 3*w^2 + 4*w - 11],
[373, 373, 2*w^3 - 5*w^2 - 5*w - 2],
[383, 383, w^3 - 5*w^2 + w + 13],
[389, 389, 3*w^3 - 10*w^2 - 4*w + 16],
[397, 397, -w^3 + 2*w^2 + 3*w + 5],
[397, 397, -3*w^3 + 7*w^2 + 10*w - 7],
[419, 419, -w^3 + 2*w^2 + 4*w + 4],
[419, 419, 3*w^3 - 7*w^2 - 12*w + 5],
[421, 421, -w^2 + 5*w - 2],
[421, 421, 3*w^3 - 7*w^2 - 10*w + 5],
[431, 431, w^2 - 10],
[439, 439, w^3 - w^2 - 7*w - 7],
[461, 461, -2*w^3 + 7*w^2 + 5*w - 10],
[467, 467, -3*w^3 + 6*w^2 + 14*w - 4],
[479, 479, 3*w^2 - 6*w - 5],
[479, 479, -4*w^3 + 11*w^2 + 10*w - 10],
[491, 491, -w^3 + w^2 + 6*w - 1],
[499, 499, -4*w^3 + 9*w^2 + 15*w - 10],
[499, 499, 2*w^3 - 5*w^2 - 7*w + 2],
[503, 503, -2*w^3 + 7*w^2 - 8],
[521, 521, 2*w^2 - w - 7],
[521, 521, -3*w^3 + 7*w^2 + 11*w - 11],
[569, 569, 2*w^2 - 5*w - 1],
[569, 569, -4*w - 1],
[571, 571, -w^3 + 3*w^2 - 7],
[577, 577, -w^3 + w^2 + 8*w - 4],
[587, 587, -3*w^3 + 8*w^2 + 7*w - 8],
[593, 593, 3*w^2 - 4*w - 11],
[593, 593, -3*w^3 + 9*w^2 + 6*w - 14],
[601, 601, -3*w^3 + 9*w^2 + 4*w - 10],
[601, 601, 5*w^3 - 9*w^2 - 24*w - 2],
[607, 607, 2*w^2 - 7*w - 5],
[613, 613, w^3 - 6*w - 2],
[613, 613, -w^2 + 3*w + 8],
[617, 617, -w^2 + 4*w - 5],
[619, 619, 3*w^3 - 6*w^2 - 13*w + 5],
[625, 5, -5],
[631, 631, -2*w^3 + 4*w^2 + 8*w - 7],
[647, 647, -2*w^3 + 4*w^2 + 5*w - 1],
[647, 647, w^3 - 3*w^2 - 2*w - 2],
[653, 653, w^3 - 4*w^2 - w + 13],
[653, 653, w^3 - 2*w^2 - 7*w + 5],
[659, 659, -4*w^3 + 10*w^2 + 15*w - 14],
[659, 659, 5*w^3 - 12*w^2 - 17*w + 14],
[661, 661, w^2 - 5*w - 2],
[661, 661, -w^3 + 5*w^2 - 2*w - 7],
[673, 673, -w^3 + w^2 + 6*w - 2],
[683, 683, 2*w^3 - 8*w^2 + w + 8],
[683, 683, 5*w^3 - 12*w^2 - 19*w + 14],
[701, 701, -2*w^3 + 6*w^2 + 5*w - 4],
[709, 709, -w^2 + 3*w - 4],
[709, 709, -2*w^3 + 5*w^2 + 9*w - 8],
[719, 719, w^2 - 7],
[719, 719, -2*w^3 + 3*w^2 + 8*w + 5],
[733, 733, -2*w^3 + 7*w^2 + 3*w - 10],
[739, 739, 4*w^3 - 11*w^2 - 13*w + 14],
[739, 739, -5*w^3 + 10*w^2 + 21*w - 4],
[739, 739, 3*w^3 - 8*w^2 - 6*w + 8],
[739, 739, -w^3 + w^2 + 8*w - 5],
[743, 743, -w - 5],
[743, 743, w^3 - w^2 - 8*w - 8],
[751, 751, -2*w^3 + 5*w^2 + 8*w - 2],
[751, 751, -2*w^3 + 6*w^2 - 1],
[757, 757, -2*w^3 + 6*w^2 + 3*w - 10],
[757, 757, 5*w^2 - 9*w - 22],
[761, 761, -2*w^3 + 4*w^2 + 9*w - 7],
[761, 761, 2*w^3 - 3*w^2 - 12*w - 1],
[769, 769, 5*w^3 - 13*w^2 - 14*w + 10],
[773, 773, -3*w^3 + 5*w^2 + 16*w + 1],
[773, 773, w^3 - 11*w + 2],
[787, 787, -w^3 + 4*w^2 - w - 11],
[797, 797, -5*w^3 + 12*w^2 + 17*w - 11],
[809, 809, -2*w^3 + 7*w^2 + 4*w - 11],
[809, 809, 4*w^3 - 9*w^2 - 14*w + 4],
[821, 821, -w^3 + 5*w^2 - 2*w - 17],
[827, 827, -3*w^3 + 8*w^2 + 9*w - 7],
[829, 829, 2*w^3 - 6*w^2 - 5*w + 2],
[829, 829, -w^3 + 4*w^2 - w - 10],
[839, 839, w^2 - 8],
[853, 853, -5*w^3 + 12*w^2 + 18*w - 16],
[853, 853, w^3 - 6*w - 8],
[863, 863, 5*w^3 - 11*w^2 - 19*w + 7],
[881, 881, -3*w^3 + 9*w^2 + 9*w - 11],
[907, 907, -w^3 + 2*w^2 + 7*w - 4],
[907, 907, -2*w^3 + 3*w^2 + 7*w + 4],
[929, 929, -w^3 + 5*w^2 - 3*w - 11],
[937, 937, 2*w^3 - 5*w^2 - 6*w + 11],
[937, 937, -2*w^3 + 6*w^2 + 3*w - 11],
[941, 941, -3*w^3 + 8*w^2 + 11*w - 8],
[953, 953, -4*w^3 + 10*w^2 + 10*w - 13],
[953, 953, w^3 - 5*w^2 - w + 13],
[967, 967, 3*w^3 - 7*w^2 - 9*w + 7],
[967, 967, -2*w^3 + 5*w^2 + 8*w + 1],
[967, 967, 4*w^3 - 13*w^2 - 6*w + 20],
[967, 967, -2*w^2 + 3*w + 13],
[971, 971, -6*w^3 + 13*w^2 + 25*w - 8],
[977, 977, w^3 - w^2 - 4*w - 10],
[977, 977, 2*w^3 - 6*w^2 - 7*w + 13],
[983, 983, -4*w^3 + 10*w^2 + 11*w - 13],
[983, 983, -w^3 + 5*w^2 - w - 7],
[983, 983, 3*w^3 - 9*w^2 - 7*w + 19],
[983, 983, -2*w^3 + 3*w^2 + 10*w - 1],
[997, 997, 5*w^3 - 12*w^2 - 17*w + 13],
[997, 997, -2*w^3 + 3*w^2 + 13*w - 4],
[997, 997, w^3 - 7*w - 2],
[997, 997, w^3 - 9*w - 2]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^8 - 21*x^6 + 134*x^4 - 288*x^2 + 196;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 2/7*e^7 - 11/2*e^5 + 403/14*e^3 - 233/7*e, 1/2*e^6 - 19/2*e^4 + 49*e^2 - 55, e, -1, -3/14*e^7 + 4*e^5 - 283/14*e^3 + 166/7*e, 1/2*e^6 - 19/2*e^4 + 49*e^2 - 56, 1/7*e^7 - 3*e^5 + 127/7*e^3 - 190/7*e, 5/14*e^7 - 7*e^5 + 523/14*e^3 - 293/7*e, -4/7*e^7 + 11*e^5 - 403/7*e^3 + 459/7*e, 1/14*e^7 - 3/2*e^5 + 67/7*e^3 - 130/7*e, -1/2*e^7 + 19/2*e^5 - 49*e^3 + 58*e, 5/7*e^7 - 14*e^5 + 530/7*e^3 - 649/7*e, 1/2*e^6 - 19/2*e^4 + 48*e^2 - 48, 1/2*e^6 - 19/2*e^4 + 47*e^2 - 50, e^4 - 12*e^2 + 21, 13/14*e^7 - 18*e^5 + 1329/14*e^3 - 787/7*e, -2/7*e^7 + 11/2*e^5 - 403/14*e^3 + 233/7*e, -1/2*e^6 + 19/2*e^4 - 50*e^2 + 53, -9/14*e^7 + 25/2*e^5 - 456/7*e^3 + 470/7*e, -e^6 + 19*e^4 - 99*e^2 + 121, 4/7*e^7 - 11*e^5 + 396/7*e^3 - 417/7*e, -1/2*e^6 + 19/2*e^4 - 52*e^2 + 78, -6, -9/14*e^7 + 25/2*e^5 - 463/7*e^3 + 533/7*e, -3/14*e^7 + 4*e^5 - 269/14*e^3 + 96/7*e, e^4 - 12*e^2 + 30, -8/7*e^7 + 22*e^5 - 806/7*e^3 + 939/7*e, -1/2*e^6 + 19/2*e^4 - 51*e^2 + 73, e^6 - 19*e^4 + 97*e^2 - 99, 5/14*e^7 - 13/2*e^5 + 209/7*e^3 - 146/7*e, -e^4 + 13*e^2 - 22, 4, -1/2*e^7 + 19/2*e^5 - 50*e^3 + 66*e, 1/14*e^7 - e^5 + 15/14*e^3 + 108/7*e, -17/14*e^7 + 47/2*e^5 - 859/7*e^3 + 943/7*e, -23/14*e^7 + 32*e^5 - 2375/14*e^3 + 1373/7*e, -e^4 + 13*e^2 - 32, -17/14*e^7 + 47/2*e^5 - 873/7*e^3 + 1027/7*e, e^6 - 20*e^4 + 113*e^2 - 141, 11/14*e^7 - 15*e^5 + 1103/14*e^3 - 688/7*e, 2*e^6 - 37*e^4 + 183*e^2 - 196, 1/2*e^6 - 19/2*e^4 + 52*e^2 - 77, 19/14*e^7 - 53/2*e^5 + 1000/7*e^3 - 1259/7*e, -27/14*e^7 + 75/2*e^5 - 1389/7*e^3 + 1613/7*e, -2*e^6 + 38*e^4 - 195*e^2 + 236, 23/14*e^7 - 63/2*e^5 + 1142/7*e^3 - 1310/7*e, 3/14*e^7 - 9/2*e^5 + 173/7*e^3 - 145/7*e, -5/2*e^7 + 97/2*e^5 - 256*e^3 + 299*e, -5/2*e^6 + 99/2*e^4 - 268*e^2 + 312, -7/2*e^6 + 135/2*e^4 - 355*e^2 + 419, -3/2*e^6 + 61/2*e^4 - 172*e^2 + 210, e^6 - 18*e^4 + 85*e^2 - 88, -2*e^6 + 40*e^4 - 219*e^2 + 257, 2*e^6 - 38*e^4 + 197*e^2 - 237, -1/7*e^7 + 2*e^5 - 22/7*e^3 - 97/7*e, -e^6 + 19*e^4 - 99*e^2 + 123, 3*e^2 - 20, -8/7*e^7 + 22*e^5 - 813/7*e^3 + 988/7*e, -3/7*e^7 + 9*e^5 - 367/7*e^3 + 472/7*e, 18/7*e^7 - 99/2*e^5 + 3627/14*e^3 - 2160/7*e, e^4 - 9*e^2 + 4, 3/2*e^6 - 59/2*e^4 + 156*e^2 - 176, -1/2*e^6 + 21/2*e^4 - 60*e^2 + 84, -4/7*e^7 + 11*e^5 - 396/7*e^3 + 375/7*e, -3/7*e^7 + 17/2*e^5 - 685/14*e^3 + 556/7*e, -1/7*e^7 + 3*e^5 - 113/7*e^3 + 85/7*e, 5/2*e^6 - 97/2*e^4 + 257*e^2 - 288, -5/2*e^6 + 99/2*e^4 - 270*e^2 + 334, -19/14*e^7 + 53/2*e^5 - 993/7*e^3 + 1238/7*e, -3/2*e^6 + 57/2*e^4 - 141*e^2 + 141, 5/2*e^6 - 97/2*e^4 + 252*e^2 - 277, 3/2*e^6 - 57/2*e^4 + 144*e^2 - 166, -3*e^6 + 58*e^4 - 305*e^2 + 360, 1/2*e^6 - 21/2*e^4 + 60*e^2 - 76, 13/14*e^7 - 37/2*e^5 + 703/7*e^3 - 829/7*e, -3/2*e^6 + 61/2*e^4 - 171*e^2 + 205, 3/14*e^7 - 7/2*e^5 + 82/7*e^3 + 9/7*e, -9/7*e^7 + 24*e^5 - 835/7*e^3 + 856/7*e, 1/14*e^7 - 5/2*e^5 + 165/7*e^3 - 431/7*e, -5/2*e^6 + 97/2*e^4 - 261*e^2 + 328, -5/2*e^6 + 101/2*e^4 - 279*e^2 + 328, e^6 - 20*e^4 + 106*e^2 - 107, 8/7*e^7 - 45/2*e^5 + 1703/14*e^3 - 1051/7*e, 3/7*e^7 - 15/2*e^5 + 475/14*e^3 - 206/7*e, 9/14*e^7 - 25/2*e^5 + 456/7*e^3 - 463/7*e, -1/2*e^7 + 19/2*e^5 - 48*e^3 + 47*e, 9/7*e^7 - 24*e^5 + 828/7*e^3 - 800/7*e, 2/7*e^7 - 6*e^5 + 233/7*e^3 - 184/7*e, 2*e^6 - 39*e^4 + 203*e^2 - 220, -1/2*e^6 + 17/2*e^4 - 35*e^2 + 20, 39/14*e^7 - 107/2*e^5 + 1948/7*e^3 - 2256/7*e, 47/14*e^7 - 131/2*e^5 + 2449/7*e^3 - 2925/7*e, 27/14*e^7 - 37*e^5 + 2659/14*e^3 - 1403/7*e, 5/2*e^6 - 95/2*e^4 + 245*e^2 - 268, -19/7*e^7 + 52*e^5 - 1895/7*e^3 + 2154/7*e, 1/2*e^7 - 10*e^5 + 109/2*e^3 - 59*e, 1/2*e^6 - 21/2*e^4 + 55*e^2 - 22, -5*e^6 + 99*e^4 - 536*e^2 + 638, -3/2*e^6 + 55/2*e^4 - 135*e^2 + 138, -11/7*e^7 + 59/2*e^5 - 2073/14*e^3 + 1061/7*e, e^7 - 19*e^5 + 95*e^3 - 97*e, -17/14*e^7 + 47/2*e^5 - 866/7*e^3 + 1097/7*e, e^6 - 17*e^4 + 75*e^2 - 70, 1/2*e^6 - 19/2*e^4 + 48*e^2 - 44, 33/14*e^7 - 46*e^5 + 3477/14*e^3 - 2183/7*e, 9/2*e^6 - 175/2*e^4 + 464*e^2 - 536, 1/2*e^7 - 10*e^5 + 113/2*e^3 - 84*e, -10/7*e^7 + 28*e^5 - 1060/7*e^3 + 1256/7*e, -1/2*e^6 + 17/2*e^4 - 35*e^2 + 37, -3/2*e^6 + 55/2*e^4 - 131*e^2 + 142, -27/14*e^7 + 75/2*e^5 - 1417/7*e^3 + 1823/7*e, 55/14*e^7 - 153/2*e^5 + 2852/7*e^3 - 3405/7*e, 16/7*e^7 - 89/2*e^5 + 3301/14*e^3 - 1836/7*e, 23/7*e^7 - 127/2*e^5 + 4687/14*e^3 - 2809/7*e, -3*e^6 + 57*e^4 - 291*e^2 + 310, 9/14*e^7 - 13*e^5 + 1031/14*e^3 - 610/7*e, -1/2*e^6 + 23/2*e^4 - 73*e^2 + 83, 37/14*e^7 - 103/2*e^5 + 1926/7*e^3 - 2318/7*e, 11/7*e^7 - 31*e^5 + 1180/7*e^3 - 1509/7*e, -4*e^6 + 78*e^4 - 420*e^2 + 502, 3/2*e^7 - 29*e^5 + 307/2*e^3 - 198*e, -2*e^2 + 43, 7/2*e^6 - 133/2*e^4 + 342*e^2 - 394, 3/2*e^6 - 59/2*e^4 + 160*e^2 - 198, 15/7*e^7 - 41*e^5 + 1492/7*e^3 - 1751/7*e, -8/7*e^7 + 22*e^5 - 813/7*e^3 + 1058/7*e, 5/7*e^7 - 13*e^5 + 453/7*e^3 - 586/7*e, -17/14*e^7 + 24*e^5 - 1851/14*e^3 + 1216/7*e, -2*e^6 + 36*e^4 - 172*e^2 + 177, -4*e^6 + 80*e^4 - 438*e^2 + 514, 19/7*e^7 - 105/2*e^5 + 3881/14*e^3 - 2231/7*e, 3*e^6 - 57*e^4 + 285*e^2 - 304, 3*e^6 - 56*e^4 + 288*e^2 - 358, -9/2*e^6 + 177/2*e^4 - 474*e^2 + 555, -1/2*e^7 + 21/2*e^5 - 65*e^3 + 108*e, -7/2*e^7 + 135/2*e^5 - 356*e^3 + 423*e, -6/7*e^7 + 17*e^5 - 636/7*e^3 + 685/7*e, e^7 - 39/2*e^5 + 207/2*e^3 - 122*e, -2/7*e^7 + 6*e^5 - 254/7*e^3 + 345/7*e, -11/2*e^6 + 211/2*e^4 - 550*e^2 + 629, 13/7*e^7 - 71/2*e^5 + 2581/14*e^3 - 1504/7*e, -3*e^6 + 57*e^4 - 293*e^2 + 345, 15/14*e^7 - 39/2*e^5 + 641/7*e^3 - 557/7*e, -4*e^6 + 77*e^4 - 406*e^2 + 480, -5/14*e^7 + 7*e^5 - 537/14*e^3 + 328/7*e, -26/7*e^7 + 71*e^5 - 2581/7*e^3 + 2987/7*e, -1/2*e^6 + 21/2*e^4 - 60*e^2 + 80, 3*e^6 - 58*e^4 + 311*e^2 - 392, 47/14*e^7 - 131/2*e^5 + 2449/7*e^3 - 2953/7*e, e^6 - 19*e^4 + 91*e^2 - 67, 15/7*e^7 - 42*e^5 + 1576/7*e^3 - 1807/7*e, 29/14*e^7 - 83/2*e^5 + 1628/7*e^3 - 2139/7*e, -6/7*e^7 + 35/2*e^5 - 1363/14*e^3 + 776/7*e, -3*e^6 + 59*e^4 - 319*e^2 + 374, -7/2*e^6 + 135/2*e^4 - 351*e^2 + 417, -e^6 + 17*e^4 - 72*e^2 + 28, 1/2*e^6 - 21/2*e^4 + 59*e^2 - 68, 1/14*e^7 - 1/2*e^5 - 38/7*e^3 + 262/7*e, -27/14*e^7 + 75/2*e^5 - 1410/7*e^3 + 1690/7*e, -7/2*e^6 + 137/2*e^4 - 368*e^2 + 425, -47/14*e^7 + 65*e^5 - 4779/14*e^3 + 2750/7*e, e^7 - 19*e^5 + 99*e^3 - 119*e, -e^4 + 17*e^2 - 46, e^6 - 19*e^4 + 100*e^2 - 148, 1/14*e^7 - 2*e^5 + 211/14*e^3 - 88/7*e, -9/2*e^6 + 175/2*e^4 - 466*e^2 + 574, -5/2*e^6 + 101/2*e^4 - 285*e^2 + 348, -1/14*e^7 + 167/14*e^3 - 276/7*e, 5/2*e^6 - 95/2*e^4 + 248*e^2 - 309, 5*e^6 - 96*e^4 + 495*e^2 - 552, 13/14*e^7 - 39/2*e^5 + 808/7*e^3 - 1123/7*e, 3/2*e^7 - 61/2*e^5 + 174*e^3 - 238*e, -5/14*e^7 + 13/2*e^5 - 209/7*e^3 + 216/7*e, 17/14*e^7 - 24*e^5 + 1809/14*e^3 - 1055/7*e, -47/14*e^7 + 129/2*e^5 - 2386/7*e^3 + 2967/7*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {13, 13, -w + 2}>] := 1;

// EXAMPLE:
// pp := Factorization(2*ZF)[1][1];
// heckeEigenvalues[pp];

print "To reconstruct the Hilbert newform f, type
  f, iso := Explode(make_newform());";

function make_newform();
 M := HilbertCuspForms(F, NN);
 S := NewSubspace(M);
 // SetVerbose("ModFrmHil", 1);
 NFD := NewformDecomposition(S);
 newforms := [* Eigenform(U) : U in NFD *];

 if #newforms eq 0 then;
  print "No Hilbert newforms at this level";
  return 0;
 end if;

 print "Testing ", #newforms, " possible newforms";
 newforms := [* f: f in newforms | IsIsomorphic(BaseField(f), K) *];
 print #newforms, " newforms have the correct Hecke field";

 if #newforms eq 0 then;
  print "No Hilbert newform found with the correct Hecke field";
  return 0;
 end if;

 autos := Automorphisms(K);
 xnewforms := [* *];
 for f in newforms do;
  if K eq RationalField() then;
   Append(~xnewforms, [* f, autos[1] *]);
  else;
   flag, iso := IsIsomorphic(K,BaseField(f));
   for a in autos do;
    Append(~xnewforms, [* f, a*iso *]);
   end for;
  end if;
 end for;
 newforms := xnewforms;

 for P in primes do;
  xnewforms := [* *];
  for f_iso in newforms do;
   f, iso := Explode(f_iso);
   if HeckeEigenvalue(f,P) eq iso(heckeEigenvalues[P]) then;
    Append(~xnewforms, f_iso);
   end if;
  end for;
  newforms := xnewforms;
  if #newforms eq 0 then;
   print "No Hilbert newform found which matches the Hecke eigenvalues";
   return 0;
  else if #newforms eq 1 then;
   print "success: unique match";
   return newforms[1];
  end if;
  end if;
 end for;
 print #newforms, "Hilbert newforms found which match the Hecke eigenvalues";
 return newforms[1];

end function;